United Nations Educational, Scientific Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS BOSON AND FERMION MANY-BODY ASSEMBLIES: FINGERPRINTS OF EXCITATIONS IN THE GROUND-STATE WAVE FUNCTIONS, WITH EXAMPLES OF SUPERFLUID 4 He AND THE HOMOGENEOUS CORRELATED ELECTRON LIQUID
After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. But because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor S(k), in the long wavelength limit. This is best understood by means of the Ornstein-Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4 He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations, and the superfluid density. Finally some future directions are pointed.
After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. But because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor S(k), in the long wavelength limit. This is best understood by means of the Ornstein-Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4 He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations, and the superfluid density. Finally some future directions are pointed.
After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. Because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor, S(k), in the long wavelength limit. This is best understood by means of the Ornstein–Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations and the superfluid density. Finally, some future directions are pointed.
In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where lnf(r) has an infinite range. The other factor is assumed to have this same form but with lnf(r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors k, the structure factor S(k) is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature T, S(k) tends to the constant value mc2kBT, where m is the mass of the particles and c the velocity of propagation of the phonons. The momentum distribution nk has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-k behavior of both S(k) and nk is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.
Abstract In this chapter we introduce some basic concepts of superfluid 4He that we will draw on in Chapters 8-12. Liquid 4 He has a long and rich history and there are many concepts that are implicit in the field today. We begin with a sketch of the unique mass and heat flow character of superfluid 4He. The assumptions of the two-fluid model used successfully to describe these properties are noted in §7.1. We introduce Bose-Einstein condensation in a Bose gas and the extension of this to interacting superfluid 4 He in §7.2. This leads naturally to the one-body density matrix which is the intermediate scattering function in the impulse approximation,We introduce a number of basic properties such as the static structure factor, the pair correlation function, and the momentum distribution which can be calculated from first principles using Monte Carlo and path integral Monte Carlo (PIMC) methods. We conclude by noting the two current descriptions of the characteristic excitation in superfluid 4He. The first is a density excitation, based on the Landau and Feynman view, with suitable modifications at higher Q to take account of correlations among atoms. In the second, the density excitation is separated into a part due to the excitation of single particles out of the condensate to states above the condensate and a part due to regular density excitations involving only states above the condensate. In the latter picture the condensate appears explicitly in S(Q, ω).
Understanding quantum many-body systems with interactions is an important problem common to various fields of physics, such as atomic physics, condensed matter physics, nuclear physics, and particle physics.Among others, ultracold atoms have provided ideal grounds to study strongly interacting systems and to explore exotic quantum states because of their high controllability.We can widely control the strengths of the interactions from weak to strong as well as the geometry of the systems.In order to deepen our understanding of quantum manybody systems with interactions, we in this thesis investigate two types of one-dimensional (1D) systems realizable with ultracold atoms; bosons and spinless fermions near two-body resonances and bosons near a three-body resonance.One-dimensional systems near two-body resonances have been intensively studied because, in the homogeneous cases, they are exactly solvable systems.The energy spectra and thermodynamic quantities exactly obtained by the Bethe ansatz have greatly contributed to deepening our comprehension of many-body systems in 1D.On the other hand, the exact computations of correlation functions, which encode information about excitations of the systems, are in general much more complicated even in the integrable systems.Motivated by this background, we derive a series of exact relations for correlation functions in bosons and fermions near two-body resonances.These relations called universal relations originate from the universal properties of the resonant systems: Their properties depend on the interactions only through the scattering lengths characterizing the low-energy scattering.The universal relations are strong constraints on the systems because the relations hold for any particle number, scattering length, temperature, and with or without a trapping potential.The relations include the asymptotic behaviors of correlation functions in high-energy regime as well as the energy relations, in which the energies are expressed in term of the momentum distributions.The universal relations involve two-and three-body contacts, which are the integrals of local pair and triad correlations, respectively.Firstly we derive, in both bosons and fermions, universal relations for static correlation functions such as static structure factors and momentum distributions within the first-quantized formalisms.The power laws of the static correlation functions for large momentum and the energy relations are obtained.The coefficients in these power-law behaviors are proportional to the two-body contact.We clarify that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one.Secondly, we derive universal relations for dynamic correlation functions for bosons.Using the operator product expansion (OPE) in the field theoretical formalism, we obtain the largeenergy-momentum behaviors of the dynamic structure factor, the dynamic current correlation, and the single-particle spectral density as well as the quasiparticle energy and the scattering rate in high-energy regime.While the behaviors of the dynamic structure factor and the current correlation are proportional to the two-body contact, that of the single-particle spectral density is proportional to the number density.Because of the interrelation between 1D bosons and fermions near two-body resonances, the results of the dynamic structure factor and the current i
We develop a systematic theory of multiparticle excitations in strongly interacting Fermi systems. Our work is the generalization of the time-honored work by Jackson, Feenberg, and Campbell for bosons, that provides, in its most advanced implementation, quantitative predictions for the dynamic structure function in the whole experimentally accessible energy/momentum regime. Our view is that the same physical effects, namely, fluctuations of the wave function at an atomic length scale are responsible for the correct energetics of the excitations in both Bose and Fermi fluids. Besides a comprehensive derivation of the fermion version of the theory and discussion of the approximations made, we present results for homogeneous 3He and electrons in three dimensions. We find indeed a significant lowering of the zero-sound mode in 3He and a broadening of the collective mode due to the coupling to two-particle--two-hole excitations in good agreement with experiments. The most visible effect in electronic systems is the appearance of a ``double-plasmon'' excitation.
In the formulation of the theory of the interacting many-boson system, the sign of the Fourier Transform of the two-particle interaction in the limit of long-wavelengths plays a dominant role in determining the character of the theory. With this in mind, we have obtained the following result: If the interaction between two neutral atoms is considered in the adiabatic approximation, and if the electronic wave functions correspond to the lowest energy state, then the sign of the scattering length will be negative. For the case of negative scattering lengths, an expansion is given for the ground-state energy of a system of interacting Bose particles. The expansion is performed in a way that eliminates the third-order divergence which results when the entire interaction is used as the perturbation. The structure factor, computed to first order, is in qualitative agreement with experiment, for large momentum values. For non-zero temperatures, expressions are obtained for the depletion of the zero-momentum mode and the specific heat. The results for the specific heat exhibit a discontinuity at the critical temperature. Quantitative estimates for the magnitude of the specific heat at the critical point yield results which are in fair agreement with empirical values for liquid helium. (Author)